The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given. Let V be a projective algebraic manifold. Methods of quantum field theory recently led to a prediction of some numerical characteristics of the space of algebraic curves in V, especially of genus zero, eventually endowed with a parametrization and marked points. It turned out that

We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick's theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

to Liza Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the Calogero-Moser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich’s formalism of ‘non-commutative Symplectic geometry’. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of P-geometry, a ‘non-commutative geometry ’ for an algebra over an arbitrary cyclic Koszul operad.

Abstract. We introduce the notion of a dioperad to describe certain operations with multiple inputs and multiple outputs. The framework of Koszul duality for operads is generalized to dioperads. We show that the Lie bialgebra dioperad is Koszul. The current interests in the understanding of various algebraic structures using operads is partly due to the theory of Koszul duality for operads; see eg. [K] or [L] for surveys. However, algebraic structures such as bialgebras and Lie bialgebras, which involve both multiplication and comultiplication, or bracket and cobracket, are defined using PROP’s (cf. [Ad]) rather than operads. Inspired by the theory of string topology of Chas-Sullivan ([ChS], [Ch], [Tr]), Victor Ginzburg suggested to the author that there should be a theory of Koszul duality for PROP’s. The present paper results from the observation that when the defining relations between the generators of a PROP are spanned over trees, then the ”tree-part ” of the PROP has the structure of a dioperad. We show that one can set up a theory of Koszul duality for dioperads. In §1, we give the definition of a dioperad and other generalities. In §2, we define the notion of a quadratic dioperad, its quadratic dual, and introduce our main example of Lie bialgebra dioperad. In §3, we define the cobar dual of a dioperad. A quadratic dioperad is Koszul if its cobar dual is quasi-isomorphic to its quadratic dual. The formalism in §2 and §3, in the case of operads, is due to Ginzburg-Kapranov [GiK]. In §4, we prove a proposition to be used later in §5. This proposition is a generalization of a result of Shnider-Van Osdol [SVO]. In §5, we prove that Koszulity of a quadratic dioperad is equivalent to exactness of certain Koszul complexes. In the case of operads, this is again due to Ginzburg-Kapranov, with a different proof by Shnider-Van Osdol. The Koszulity of the Lie bialgebra dioperad follows from this and an adaptation of results of Markl [M2].

(0.1) Morita theory. Let A, B be two commutative rings. If their respective categories of modules are equivalent, then A and B are isomorphic. This is not anymore true if A and/or B are not assumed to be commutative. Morita theory

Abstract. We show that the Connes-Moscovici cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree −2. More generally, we show that a ”cyclic operad with multiplication” is a cocyclic module whose cohomology is a Batalin-Vilkovisky algebra and whose cyclic cohomology is a graded Lie algebra of degree −2. This explain why the Hochschild cohomology algebra of a symmetric algebra is a Batalin-Vilkovisky algebra. 1.

We show that a graded commutative algebra A with any square zero odd differential operator is a natural generalization of the Batalin–Vilkovisky algebra. While such an operator of order 2 defines a Lie algebra structure on A, an operator of an order higher than 2 (Koszul–Akman definition) leads to the structure of a strongly homotopy Lie algebra (L∞–algebra) on A. This allows us to give a definition of a Batalin–Vilkovisky algebra up-to homotopy. We also make an important conjecture generalizing Kontsevich formality theorem to the Batalin–Vilkovisky algebra level. 1. Introduction. Batalin–Vilkovisky algebras are graded commutative algebras with an extra structure given by a second order differential operator of square 0. The simplest example is the algebra of polyvector fields on a vector space R n. There is a second order square zero differential operator on this algebra. This operator comes

Some binary quadratic operads are endowed with anticyclic structures and their characteristic functions as anticyclic operads are determined, or conjectured in one case. 0

Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in non-commutative geometries based upon these “ring–like ” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references. 0.1. Inner cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and B =